Question: A spherical scoop of vanilla ice cream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the ice cream spreads out uniformly forming a cylindrical region 8 inches in radius. Assuming the density of the ice cream remains constant, how many inches deep is the melted ice cream? Express your answer as a common fraction.
The ice cream sphere has volume $\frac{4}{3}\pi(2^3) = \frac{32\pi}{3}$ cubic inches.  Let the height of the cylindrical region be $h$; then, the volume of the cylindrical region is $\pi (8^2)h=64\pi h$.  Thus, we have \[\frac{32\pi}{3} = 64\pi h.\] Dividing both sides by $64\pi$ yields $h = \boxed{\frac{1}{6}}$ inches.